Physics Letters A 160 ( 1991 ) 1-8 North-Holland
PHYSICS LETTERS A
N-measurement Bell inequalities, N-atom entangled states, and the nonlocality of one photon Lucien Hardy Department of Mathematical Sciences, University of Durham. Durham DH1 3LE, UK Received I l June 1991; revised manuscript received 2 September 1991;accepted for publication 6 September 1991 Communicated by J.P. Vigier
The derivation of the CHSH Bell inequalities is generalized to an N-measurement scheme. It is then shown that one single photon can be used to prepare (1) an N-atom entangled state, and (2) an ensemble of N-atom entangled states. Measurements on the latter demonstrate that, in quantum theory, one single photon can exhibit nonlocality.
Bell's inequalities [ 1 ] apply to experiments in which two simultaneous measurements are made on a quantum system in distant regions. Recently, Greenberger, H o m e and Zeilinger ( G H Z ) have considered gedankenexperiments in which more than two measurements are made on a q u a n t u m system  (see also refs. [ 3 - 5 ] ). In these gedankenexperiments perfect correlations exist between the measurements and, because o f this, it is possible to derive a direct contradiction between quantum mechanics and locality without using inequalities. However, inequalities are still o f interest for two reasons: (i) to elucidate the relationship between the G H Z proof o f nonlocality and the proof o f nonlocality using Bell inequalities; (ii) the perfect correlations considered in gedankenexperiments are not possible in real experiments. Therefore, inequalities would be necessary if an experimental demonstration of nonlocality in the schemes involving more than two measurements is to be possible. Greenberger, H o m e , Shimony and Zeilinger have derived the necessary inequalities to test the G H Z contradiction in a real experiment (see section V o f ref. [ 3 ] ). Other authors have also derived N-measurement inequalities. Mermin [6 ] has derived inequalities for a state o f N spin-½ particles. These inequalities are violated by q u a n t u m theory for N > 2.
Suppes and Zanotti [7 ] have derived different inequalities for N > 4. In section 2 o f this paper we will show that a simple generalization o f the Bell inequalities due to Clauser, Horne, Shimony and Holt ( C H S H )  to an N-measurement scheme is possible. In section 3 it will be seen that, in those circumstances where there is the m a x i m u m possible violation o f these inequalities, a G H Z type contradiction follows. in a previous article [ 9 ] the author has shown that it is possible to prepare a two-atom entangled state using a M a c h - Z e h n d e r interferometer with a single photon source. In section 4 we will show that this method can be extended to prepare an N-atom entangled state using N - l M a c h - Z e h n d e r interferometers and a single photon source. These entangled states will then be shown to violate the N-measurement inequalities for all N~> 2. In order to test Bell's inequalities, it is necessary to consider an ensemble o f these entangled states and, hence, we need an ensemble o f single photons to prepare the entangled states. Therefore, strictly speaking, we are not seeing the nonlocality o f " a single photon". In section 5 we will consider some interesting arrangements where, using only one photon, it is possible to prepare a sufficient number of entangled states to get a violation of the N-measurement Bell inequalities.
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2. N - m e a s u r e m e n t
PHYSICS L E T T E R S A
To extend the Bell inequalities to N measurements we consider the simultaneous measurements Ak(ak) in the separate regions Rk for k = 1 to N. ak is a local variable, that is, it can be varied locally in the region Rk. We use the notation A~(a~), A2(a2), etc. rather than A(a), B(b), etc. to achieve greater notational simplicity. The possible values OfAk(ak) are + 1 and 0, where 0 corresponds to a null measurement (when the measurement apparatus has failed to register a result). Hence,
IAk(ak,2) I ~< 1 , where Ak(ak, 2)
is the expectation value of Ak(ak) when the quantum system has hidden variables 2. These hidden variables give a more complete description of the quantum system than is given by the quantum mechanical state vector. Note that, due to the assumption of locality, this expectation value depends only on ak and 2, not on a~ for i~k. In a deterministic theory, Ak(ak) is exactly determined by ak and 2. Therefore, in a deterministic theory with no null results ( 1 ) is replaced by
Ak(ak,2)= +_ 1 .
Ak(ak' ).~) )
I ~ (fiAk(ak,2A[l+--fiAk(a'~',2,,]) F/i=l
- -n,=,~l ( fi Ak(a~,2,)
where we have chosen the a~ and a [ so that
for all k and i (see below). From ( 1 ) we see that the quantities in square brackets in (5) are positive. Therefore, using (1) and (4), we obtain
Rearranging and letting n--,oo gives
where E(a) is shorthand for E(al, a2, ..., aN). The angle brackets, ( ) , denote the average over an infinite number of experiments. For local theories it is possible to write
1 ~(~=, E , ( a ) = -ni=l
1 ~(fiAk(ak,2i)-- fiAk(a'k,2,)) f/i=l
Inequalities (8) together with eq. (6) are N-measurement Bell inequalities. Eq. (6) defines the possible relationships between ak, ak; a'~ and ak". Using (6) we find that, for each value of k, there are the following possibilities,
We consider the correlation coefficient
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where E,(a)--,E(a) as n - - , ~ and 2~ are the hidden variables for the ith quantum system. Now
In the case of a deterministic theory with no null resuits (2) holds so that we have the additional possibility,
~ a k •
To obtain the two-measurement Bell inequalities from the N-measurement inequalities we put k = 1, 2. From (9a) we can put
and from (9b) we can put
Substituting these into (8) gives
Volume 160, number 1 -2<.E(a, b)-E(a,
PHYSICS LETTERSA b')+E(a', b)+E(a', b')<~2.
(10) These are the well known CHSH Bell inequalities [ 8 ] (see also ref. [ 10 ] ).
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m ~- a 2 = X tit
a 3=a 3 =x
a2 =a2" =Y,
a 3= a 3 = y ,
in inequalities (8) gives the three-measurement inequalities 3. Bell's theorem without inequalities
- 2 <,E(x, y, y ) - E ( x ,
Greenberger, Home and Zeilinger ( G H Z ) have demonstrated Bell's theorem without inequalities by considering situations in which there exist perfect correlations in schemes involving three or more measurements. In the simplest case they consider the three spin-½ particle state 1 [~)=~(l+)ll+)2l+)3 -I-),1-)21-)3),
where I -+ )j represents the state of particle j with spin + ½along the z-axis. We now consider measurements of spin along the x and y axes. Quantum mechanics predicts E(x,y,y) E(x,x,x)=
= ( e xle y 2a y 3) = + 1 ,
t 2 3) = - 1 , ( trxtrxex
= ( e ~ etx e y2)
= + 1 ,
= ( tre1try2 trx3 ) = + 1 ,
where trJx and a¢ are the spin operators of particle j for spin along the x and y axes respectively. These are perfect correlations and, therefore, for each 2, we have
x, x ) + E ( y , x, y )
The upper bound of (14) can also be obtained by putting N = 3 in Mermin's inequalities . Since IEI < l, the values ( 12a)- (12d) yield the maximum possible violation of inequality (14). In this case, the maximum violation of the Bell inequalities corresponds to the direct contradiction between quantum mechanics and locality considered above. In fact, in those circumstances where a maximum violation of the Bell inequalities can be obtained, such a contradiction will always exist. To prove this, first we note that we get a maximum violation of the Bell inequalities (8) if, and only if, E(a)=E(a")=E(a")=-E(a')=+l
Now, from (10) we have N
E ( a ) = I-I Ak(ak, 2 i ) = + 1 ,
I-I Ak(a~,2,)=-T- 1,
E ( a " ) = I-I A k ( a ~ , 2i) = + 1 ,
I-I Ak~,.,kt.... , 2 i ) = +1_ .
A t (X, ,~i)A2(Y, 2 , ) A a ( y , 2,) = + 1 ,
~4t (x, 2 i ) A 2 ( x , 2 i ) A a ( x , ,'~i) = - 1 ,
At (y, 2 i ) A 2 ( x , 2/)A3(y, 2/) = + 1 ,
At (y, .~.i)A2(y, 2 i ) A 3 ( x , }ti) = + 1 .
It is not necessary to sum over 2, when there is a perfect correlation. Taking the product of eqs. ( 1 6 a ) (16d) and using (6) gives
Now, taking the product of eqs. ( 13a)- (13d) gives a positive quantity on the l.h.s, and a negative quantity on the r.h.s. This contradiction demonstrates that quantum mechanics is nonlocal without using inequalities. It is clear from ( 13a)-(13d) that (2) and, therefore, (9c) hold. Putting N = 3 and, using ( 9 a ) - ( 9 c )
l-I [Ak(ak, 2,)Ak(a~', 2,) ]z= _ 1 .
The l.h.s, of (17) must be positive but the r.h.s, is negative, therefore we have a contradiction.
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4. Using single photons to prepare N-atom states
We take two identical spin-½ atoms, 1 and 2, and prepare them in the states
The method of preparing a two-atom entangled state using a Mach-Zehnder interferometer with a single photon source is discussed in ref. [ 9 ]. Here we will show that it is possible to extend this approach to prepare an N-atom entangled state using N - 1 interferometers and a single photon source. These entangled states will be used to illustrate the violation of the N-measurement Bell inequalities by the predictions of quantum mechanics. First, we show how to obtain a two-atom entangled state. (This is considered in greater detail in ref. .) One atom is placed in each arm of a Mach-Zehnder interferometer (see fig. 1 ). The interferometer is arranged so that, when there are no atoms in its arms, no photons will arrive at D2 due to destructive interference (see ref. [ 11 ] ). Now, a single photon is sent into the interferometer. The state of the photon going to the right is represented by Ir > and the state of the photon going up is represented by l u >. The operations of the beamsplitters on the states of the photons are
I~'k> =ot~- I + >k +a~- I - >k,
for k = 1, 2. The states I + )k have spin+ ½along the z-axis. Now, applying a magnetic field along the zaxis will cause Zeeman splitting. We choose the frequency of the single photons to correspond to some excitation energy of the I + > k states, but not to any excitation energy of the I - )k states. Hence, atoms in the states I + > k will absorb some of the photons whereas atoms in the states I - > k will be transparent to the photons. Therefore, the operations of the atoms on the state of the photon are lu) I + )k--,/tlu) I + >k+/t'10> le>~,
In> I - >k--' lu>l - >k,
where 10> indicates that the photon has been absorbed and Ie > k is the excited state of the atom when it has absorbed the photon. The initial state of the system is I ~ > = [r> I~u, > 1~2 > .
(Ir> + i l u > ) ,
(lu> + i l r > ) ,
The final state will be of the form I ~F> = It> 1~2> + lU> I~0y2>+ 10> 1~°2>,
and the operations of the mirrors are
where 1~2 >, [ ~ 2 > and [tp°2 > are state vectors in the product space of atoms 1 and 2. If we consider only those events for which a photon is detected at D2 then the state vector will collapse onto the second term on the r.h.s, of (24). The unnormalized state vector of the atoms is then given by I~Y2>. Using ( 1 8 ) - ( 2 4 ) we find that 19Y2 > = ( I - P ) ( c ~ + a~-] + >, 1 - > 2 - a i - a + I - >, l + >z) •
atom 2 ( )
atom 1 (
Fig. 1. Mach-Zehnder interferometer arranged to prepare a twoatom entangled state.
If the atoms are prepared so that they have spin + ½ along the x-axis then I~'k>= ~
(1+ >k+l-->k) •
Therefore, a~" =a~- = 1/x/~. Hence, when the photon is detected at Dz the normalized state vector of the atoms is, from (25), 1 I ~ > = ~7~ ( 1 + > 1 1 - > z - 1 - > , I + > z ) .
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This is a singlet state and it is well known that suitable measurements on a singlet state will violate the Bell inequalities (10). To produce a three-atom entangled state we can use the arrangement shown in fig. 2. This arrangement consists of two Mach-Zehnder interferometers placed so that the "dark" output of the first, MZI, goes into the input of the second, MZII. Furthermore, one arm of each interferometer goes through atom 2 so that MZI has the atoms 1 and 2, one in each arm, and MZII has the atoms 2 and 3, one in each arm. We shall refer to two interferometers when arranged in this way as entangled. Now, if the photon emerges at output E,, then from (25) the unnormalised state vector of the atoms is I(Pl2 > = (--oq-a~- I -- ) , ) [ + >2 +(a~ot~- I + >,)1-- >2-
The photon now goes into MZII. The state of atom 2 is now given by (28) and the state of atom 3 is
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a?--,z-a~ a~ I-v->,, a~-.a~. The photon could now be fed into a third interferometer, MZIII, entangled with MZII such that atom 3 is in an arm of both MZII and MZIII. We would then have a four-atom entangled state. In principle, we can have as many entangled interferometers as we wish. Preparing all the atoms in the initial state (26) so that ot~ = l / v / 2 , we get the normalised Natom state, 1
[~'>u= ~ (1 +-- +--'-->N - ( - 1 ) u l - +--+...>N,
by using N - 1 entangled interferometers. In the notation used in (31), + in the /cth position corresponds to a spin of + ½ along the z-axis for the kth atom. Consider, now, the correlation coefficient N
E(a)=N(~°I 1-I trk(ak) I~O>N,
I W3> =ot~" I + >3 +otj- I - >3-
If the photon now emerges at the "dark" output E2 then the states of atoms 2 and 3 will become entangled. As the states of atoms 1 and 2 are already entangled we find that we have the three-atom entangled state,
1~,23> = - - a + o t ~ a ~ l + > l I - > 2 1 +>3 --oti-a~-a~- I -- >,l + >21-- >3.
We obtain (30) from (25) by making the changes 1+>.--,1+>2,
where ak(ak) is the spin operator corresponding to a measurement of spin on atom k along a direction in the xy-plane at an angle ak to the x-axis. Using elementary quantum mechanics (for example, see appendix F of ref. [ 3 ] ), we can show that
I -1- >2--i. [ "F > 3 ,
and, comparing (28) and (29) with (20),
It is well established that the Bell inequalities are violated by the predictions of quantum theory for N = 2. We will now show that, according to quantum thepry, appropriate measurements on the state (31 ) will give a maximum violation of the N-measurement Bell inequalities for all N>_-3. Using ( 9 a ) - ( 9 c ) we put al=a]=0
. .= 0 . and . . a3 = a 3 = n / 2 a3. =a3 and, for all k > 3,
ak=a'k=akvt =akcH = 0 . Using (8) we obtain the Bell inequalities Fig. 2. Two entangled Mach-Zehnder interferometersarranged to prepare a three-atom entangled state.
Volume 160, number 1 -2~E(0,-x/2,
PHYSICS LETTERSA x/2,0,0,...,0)
results. It is clear that the derivation of inequalities ( 8 ) in section 2 still holds when we replace Ak (ak, 2 i) by A ~(ak, 2,) where, now, the correlation coefficient is
- E ( 0 , 0 , 0 , 0 , 0 ..... 0)
+E(x/2,0, x/2,0, O,...,O) +E(x/2,-n/2,0,0,0
.... , 0 ) ~ 2 .
From (33) we get
But, from (4) and (36),
n / 2 , 0 , 0 , . . . , 0 ) = ( - 1 ) ~,
E ( 0 , 0 , 0 , 0 , 0 ..... 0 ) = - ( - 1 )
E(x/2,0, x/2,0, O..... 0 ) = ( - 1 ) E ( n / 2 , - ~ / 2 , 0 , 0 , 0 ..... 0 ) = ( - 1 )
E'~(a) --E.(a) +
These give a maximum violation of the Bell inequalities (34). Therefore, we can have a maximum violation of the Bell inequalities for all N>~ 3. The contradiction (17) can be demonstrated whenever there is a maximum violation of Bell's inequalities and, therefore, can be demonstrated, by using these N-atom states, for all N>~ 3. To obtain the inequality (34) (and (14) of section 3 ) required the use of the possibility (9c). Unfortunately, (9c) only applies in the case of a deterministic theory with no null results (i.e. when eq. (2) holds). However, as pointed out by Greenberger, Horne, Shimony and Zeilinger , there exist equivalence theorems (see refs. [ 12,13 ] ) showing that the predictions of any stochastic local theory can be duplicated by a deterministic local theory. Therefore, if the results of an experiment, or the predictions of quantum mechanics cannot be explained by a deterministic local theory then they cannot be explained by any local theory. Hence, inequalities obtained from (8) using possibility (9c) apply to all local theories with no null results. In a real experiment there will be null results, so we still do not have experimentally testable inequalities. However, a simple mathematical trick can be used to remove the requirement that there are no null results. Consider the measurement A~(ak) for which
A~(ak, 2i) =Ak(ak, 2i) , ifAk(ak, 2i) = _ 1 , =B~,
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ifAk(ak, 2~) = 0 ,
where B~, is generated randomly to take the values + 1 and - i. For this measurement there are no null
where K~ is the set of k for which Ak(ak, 2~) =0. AS n~ the second term on the r.h.s, of (38) will vanish because B~, is generated randomly. Hence E ' (a) = E (a) and therefore, making measurements of Ak(ak) (with null results) is equivalent to making measurements of A'k(ak) (without null results) in the limit n--,o~. Hence, inequalities obtained from (8) using (9c) apply to experiments where there are null results and, consequently, can be used in real experiments.
5. The nonlocality of one photon The experiments considered above demonstrate the nonlocality of a single photon. The single photon is used to put N atoms into an entangled state. The nonlocality of a single photon was first demonstrated by Tan, Walls and Collet . In the experiment they proposed a single photon field is split at a beam splitter and each output mode impinges on a homodyme detector. Their experiment is statistical in the sense that, for the settings of local variables considered, the correlation coefficient E has modulus less than one, i.e. [E[ < 1. Hence, it is necessary to repeat the experiment a large number of times in order to find the value of E. The same is true of measurements on a two-atom singlet state produced using a Mach-Zehnder interferometer. Consequently, both experiments require that we have a large number of single photons. Therefore, strictly speaking, we have not demonstrated the nonlocality of "a single photon" but of a large number of single photons. Even in the N-atom experiments with N>~ 3, in which we can demonstrate nonlocality by considering situations where there are perfect correlations, it is still
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necessary that we perform the experiment at least four times, once with each setting of the local variables. Realistically, we must perform the experiment many more times than this to establish that we actually have perfect correlations. The nonlocality of one single photon has not yet been demonstrated. We will now see that it is possible to demonstrate the nonlocality of one single photon. First, we note a new feature of the experiments using Mach-Zehnder interferometers. The quantum system whose nonlocal properties cause the production of an entangled state, that is the photon, actually survives unchanged after it has produced the entangled state. Therefore, rather than producing a new photon, the same photon can be used again to produce another entangled state. In principle, as many entangled states as required can be produced from one photon. The apparatus for doing this would consist of a large number of interferometer systems, each consisting of N - 1 entangled Mach-Zehnder interferometers, placed in series such that, if the photon emerges out of the "dark" output of the ith system, it is inputted into the ( i + 1 )th system. The probability of a photon being detected at the "dark" output of the nth system is very small. In most cases the photon will either emerge at one of the other outputs or be absorbed by one of the atoms. However, if we consider a photon which does survive after passing through the series, then it is possible to demonstrate the nonlocality of this photon. If the photon is detected at the dark output of the nth system then the state of the atoms is I~)~v = f i I~)~v,
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ments on the state (39). The state (39) is prepared using only one photon. We have, therefore, demonstrated the nonlocality of one single photon in quantum theory.
6. Conclusions The Bell inequalities derived by Clauser et al. in 1969 can easily be generalised to an N-measurement scheme. When there is a maximum violation of these N-measurement Bell inequalities a direct contradiction between quantum mechanics and locality can be demonstrated. These contradictions, which have already been demonstrated by GHZ, do not require that we consider any inequalities. However, in a real experiment, inequalities would be necessary because we could not have perfect correlations. It is possible to produce N-atom entangled states using N - 1 entangled Mach-Zehnder interferometers. Measurements on these entangled states will violate the N-measurement Bell inequalities for all N>~ 2. For all N>~ 3 a maximum violation of the inequalities is possible. The nonlocality of one single photon can be demonstrated by placing many Mach-Zehnder interferometers in series. Finally, we note that the experiments with Mach-Zehnder interferometers can be generalised to other single particle sources. For example, neutrons could be used. Therefore in quantum theory, one single particle of any type is capable of exhibiting nonlocality.
References where, from (31 ), I~}~= ~
[I + - + - . . . > ~ ,
- ( - 1)N[ - + - +...)~v].
The superscript i labels the atoms, k = 1 to N, in the ith system. Making one measurement on each set of Natoms for i= 1 to n is equivalent to making n measurements on the same set of N atoms prepared before each measurement in the state [~a)N given in (31 ). Therefore, we can test the Bell inequalities, and hence demonstrate nonlocality, by making measure-
[ 1] J.S. Bell, Physics 1 (1964) 195.  D.M. Greenberger, M.A. Home and A. Zeilinger, in: Bell's theorem, quantum theory,and conceptionsof the universe, ed. M. Kafatos (Kluwer, Dordrecht, 1989) p. 74. D.M. Greenberger, M.A. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 (1990) 1131.  N.D. Mermin, Phys. Rev. Lett. 65 (1990) 3373.  H.P. Stapp, Interpretation of an experiment of the Greenberger-Horne-Zeilinger kind, Lawrence Berkeley Laboratory,preprint LBL-293377(1990).  N.D. Mermin, Phys. Rev. Lett. 65 (1990) 1838. [7l P. Suppesand M. Zanotti, Found. Phys. Lett. 4 ( 1991) 101. [8 ] J.F. Clauser, M.A. Home, A. Shimonyand R.A. Holt, Phys. Rev. Len. 23 (1969) 880.
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 L. Hardy, Using a Mach-Zehnder interferometer to test Bell's inequalities, submitted to Proc. 2nd. Int. Wigner Symp. ( 1991 ).  J.S. Bell, in: Foundations of quantum mechanics, ed. B. d'Espagnat (Academic Press, New York, 1971 ) p. 171. [I1]A.C. Elitzur and L. Vaidman, Quantum mechanical interaction-free measurements, Tel Aviv preprint, submitted to Phys. Rev. Lett.( 1991 ).
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H.P. Stapp, Found. Phys. 10 (1980) 767.  A. Fine, J. Math. Phys. 23 (1982)1306. S.M. Tan, D.F. WalIs and M.J. Collett, Phys. Rev. Lett. 66 (1991) 252.